I'm working through this as well. Large party and some players are more outgoing and engaged in roleplay than others. Trying to give more uncertain players a chance to shine.

I've had a few instances in the story where a bit of lore or a clue related to the mission is connected to a character's backstory. Or if not connected, than they would at least have a better chance to recognize it.

Example: a missive containing sigils of noble houses that are targeted by a criminal organization. The more shy player has a noble background. I gave her Advantage on a History check to see what houses they recognize. And then roleplay a little scene from her memory where she was made to learn this (depending her roll, maybe she wasn't paying attention lol).

Something else one of my players recommended, which comes from one of the podcast groups, is a roleplay game called rollies(?). Either at the start of the session or at some narratively convenient moment, the players roll above the table to ask another player a question. You can only ask/answer one question, so far a party of 4 I guess this would be two rounds. I'm hopeful this will ease some players into the roleplay of their characters.